Without actual division , show that each of the following rational numbers is non -terminating repeating decimal : \frac{77}{210}

Foundation

Mathematics Foundation

Properties of Real Number

Recurring Decimal Number

Rational Numbers

Question

Without actual division , show that each of the following rational numbers is non -terminating repeating decimal :

\frac{77}{210}

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\frac{77}{210} = \frac{77}{3 \times 5 \times 2 \times 7} = \frac{11}{3 \times 5 \times 2}

2, 3 or 5 is not factor of 11 so given fraction is in its simplest form.
and,

(2 \times 3 \times 5) \neq (2^{m} \times 5^{n})

hence given fraction is non -terminating repeating decimal.